Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 219
The equation tan (0 + a) = tan 0
reduces to tan a (1 + tan2 0) = 0,
and when tan =- 1, then 0 is of the form a + if with 3 infinite.
Page 210 ab Euclide incoepti, etc.] Newton has in mind the words of Descartes
in La Géométrie, "commencée à resoudre par Euclide et poursuivie par Apollonius, sans
avoir été achevée par personne." Apollonius has indeed nothing to say about a locus
related to more than four lines, but there is no reason to question his statement that
he had solved the problem of the four-line locus. Its complete working out would have
supplied ample materials for a book on the scale of his lib. v. on Normals*.
Newton assumes Lemma xvII. in Lemma xx., on which his Lemma xxI. depends,
thus making the "Organic Description" of conics seem less simple than it is. Having
proved Prop. A, make A, B, K, P, C fixed points and D variable, and we have at
once RT parallel to the fixed line CK (p. 206) as in Lemma xxI.
Page 216 Sed propero ad magis utilia] The Principia, all but some ten or twelve
propositions composed previously, having been written in less than a year and a half
(Dec. 1684-May 1686), Newton could not have had much time to spare for the two
sections (lib. I. 4-5) on Inventio Orbiztn. Maclaurin's constructions of a conic by means
of three (p. 207, Cor. 6-7) or more lines through fixed points grew out of a lenma
NVeutonianum, as we learn from the preface to Simson's Sectiones Conicce. Newton himself,
vith leisure, could have developed the said two sections into a comprehensive and
essentially modern treatise.
* Of this lib. v. Chasles tells us that it treats of "les maximis et minimis, sur les sections coniques. Dans le
questions de maxima et de minima," and that, "On y cinquième Apollonius examine particulièrement quelles sont
retrouve tout ce que les méthodes analytiques d'aujourd'hui les plus grandes et les moindres lignes qu'on puisse tirer de
nous apprennent sur ce sujet." This astonishing statement chaque point donné à leur circonférence. On y retrouve
is a too brief summary of the words of Montucla on lib. v. tout ce que nos méthodes analytiques d'aujourd'hui nous
and lib. vI., "Ils traitent l'un et l'autre un des sujets les apprennent sur ce sujet." Chasles goes on to speak of
plus difficiles de la géométrie, savoir les questions de normals as the subject of lib. v.
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